Higher order ODE with applications

Higher Order Differential Equation&

Its Applications

Contents

IntroductionSecond Order Homogeneous DEDifferential Operators with constant coefficients

Case I: Two real roots Case II: A real double root Case III: Complex conjugate roots

Non Homogeneous Differential EquationsGeneral SolutionMethod of Undetermined CoefficientsReduction of OrderEuler-Cauchy EquationApplications

Introduction

A differential equation is an equation which contains the derivatives of a variable, such as the equation

Here x is the variable and a, b, c and d are constants.

Types of Differential EquationsHomogeneous DE

Non Homogeneous DE

0)()()()( 011

1

1

yxadxdyxa

dxydxa

dxydxa n

n

nn

n

n

)()()()()( 011

1

1 xgyxadxdyxa

dxydxa

dxydxa n

n

nn

n

n

Second Order Homogeneous DE

A linear second order homogeneous differential equation involves terms up to the second derivative of a function. For the case of constant multipliers, The equation is of the form

and can be solved by the substitution

Solution

The solution which fits a specific physical situation is obtained by substituting the solution into the equation and evaluating the various constants by forcing the solution to fit the physical boundary conditions of the problem at hand. Substituting gives

which leads to a variety of solutions, depending on the values of a and b. In physical problems, the boundary conditions determine the values of a and b, and the solution to the quadratic equation for λ reveals the nature of the solution.

Case I: Two real roots

For values of a and b such that

• there are two real roots m1 and m2 which lead to a general solution of the form

211 2( ) nm x m xm x

ny x c e c e c e

Case II: A real double root

If a and b are such that

then there is a double root λ =-a/2 and the unique form of the general solution is

Case III: Complex conjugate roots

For values of a and b such that

there are two complex conjugate roots of the form and the general solution is

The general solution of the non homogeneous differential equation

There are two parts of the solution:1. solution of the homogeneous part of DE

2. particular solution

( )ay by cy f x

cy

py

Non Homogeneous Differential Equations

General Solution of non-homogeneous equation is given by

represents solution of Homogeneous part represents particular solution

c py y y

General Solution

The method can be applied for the non – homogeneous differential equations , if the f(x) is of the form:

• constant

• polynomial function

•

•

• A finite sum, product of two or more functions of type (1- 4)

( )ay by cy f x

mxe

sin ,cos , sin , cos ,...x xx x e x e x

Method of Undetermined Coefficients

Reduction of Order

We know the general solution of is y = c1y1 + c2y1. Suppose y1(x) denotes a known solution of (1). We assume the other solution y2 has the form y2 = uy1.Our goal is to find a u(x) and this method is called reduction of order.

dxxy

exyydxxP

)()( 2

1

)(

12

Euler-Cauchy Equation

Form of Cauchy-Euler Equation

Method of SolutionWe try y = xm, since

)(011

11

1 xgyadxdyxa

dxydxa

dxydxa n

nn

nn

nn

n

k

kk

k dxydxa kmk

k xkmmmmxa )1()2)(1(

mk xkmmmma )1()2)(1(

Applications

Simple Harmonic Motion

Simple Pendulum

Applications (Cont.)

pressure change with altitude

Velocity Profile in fluid flow

Applications (Cont.)

vibration of springs

Electric circuits

Applications (Cont.)

Discharge of a capacitor

References

http://hyperphysics.phy-astr.gsu.edu/hbase/diff2.html#c2 http://tutorial.math.lamar.edu/Classes/DE/

IntroHigherOrder.aspx http://hyperphysics.phy-astr.gsu.edu/hbase/diff.html https://www.math.ksu.edu/~blanki/SecondOrderODE.pdf http://www.mylespaul.com/forums/showthread.php?t=266222

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